Try getting the box that runs the proxy working first before you
try the remote boxes.

Hi Roberto,
after setting it to 5 Minutes (you saw that), and after waiting for another
day (in fact it was over night) all of a sudden it works automagically. Now
I'm back to 60 Minutes, and it still works - when I come in in the morning,
I just wait for 60 minutes before I try an 'apt-get update' and 'apt-get
dist-upgrade', because most of the time we have to switch off the boxes
over night :-(

Waiting the 60 minutes in the morning is probably not necessary. Also,
I set my proxies to 240 minutes, but that is personal preference (in
my case I have an old slow machine acting as proxy).
Why do you need to shut down the machines? That seems very strange.
Is it company policy? unreliable electric power?